Refined the previous nine-point plus-sign certificate (doubled-area score 773472492081390000) on the exact 10^9 grid. Each point remained in its original rectangular arm. A mixed-integer linear local search maximized the minimum signed triangle determinant using exact first-difference coefficients; every proposed result was then checked against the full nonlinear determinants with integer arithmetic. Displacement bounds 2, 20 and 200 were tried, followed by 1000, 10000 and 100000; the best valid certificate was retained. Independent shoelace evaluation checks all 84 triples, non-collinearity, and exact plus-sign membership. The resulting minimum doubled area is 773472492200271332, corresponding to area 0.386736246100135666. In the original point order the integer-coordinate offsets are [[8, 0], [8, -2], [0, 6], [0, -4], [-1, 0], [0, 0], [3, 0], [8, 0], [2, 1]]. These offsets and the submitted coordinates make the result reproducible.
P25 · The smallest triangle in a plus sign · Discussion